Vacancy-formation energies at the (111) surface and in bulk Al, Cu, Ag, and Rh

The vacancies at the Al, Cu, Ag, and Rh (111) surface are investigated using total-energy and charge-density calculations. For comparison, results of the bulk vacancies are presented as well. In both cases the (\ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 )R30\ifmmode^\circ\else\textdegree\fi{} surface unit cell is used. The calculations apply density-functional theory together with the local-density approximation and the ab initio full-potential linear-muffin-tin-orbital method. The results compare well with known experimental data. In addition, the results are discussed in terms of a tight-binding model in the second moment approximation. It is found that among those metals which are studied here, Al has exceptionally small values for the vacancy-formation energies in the bulk and at the surface. This is related to the formation of a ${\mathit{sp}}^{2}$ bonding component in the Al bonds on the (111) surface for the case of a periodic vacancy structure with a (\ensuremath{\surd}3 \ifmmode\times\else\texttimes\fi{} \ensuremath{\surd}3 )R30\ifmmode^\circ\else\textdegree\fi{} surface unit cell.